An Excel spreadsheet to determine and depict end

Computers and Geosciences 113 (2018) 106–114

Contents lists available at ScienceDirect

Computers and Geosciences journal homepage: www.elsevier.com/locate/cageo

Research paper

Perovskite classification: An Excel spreadsheet to determine and depict end-member proportions for the perovskite- and vapnikite-subgroups of the perovskite supergroup Andrew J. Locock a, *, Roger H. Mitchell b a b

Department of Earth and Atmospheric Sciences, University of Alberta, 1-26 Earth Sciences Building, Edmonton, Alberta, T6G 2E3, Canada Department of Geology, Lakehead University, Thunder Bay, Ontario, P7B 5E1, Canada

A R T I C L E I N F O

A B S T R A C T

Keywords: Perovskite Electron microprobe Excel Guidelines Nomenclature Spreadsheet

Perovskite mineral oxides commonly exhibit extensive solid-solution, and are therefore classified on the basis of the proportions of their ideal end-members. A uniform sequence of calculation of the end-members is required if comparisons are to be made between different sets of analytical data. A Microsoft Excel spreadsheet has been programmed to assist with the classification and depiction of the minerals of the perovskite- and vapnikitesubgroups following the 2017 nomenclature of the perovskite supergroup recommended by the International Mineralogical Association (IMA). Compositional data for up to 36 elements are input into the spreadsheet as oxides in weight percent. For each analysis, the output includes the formula, the normalized proportions of 15 end-members, and the percentage of cations which cannot be assigned to those end-members. The data are automatically plotted onto the ternary and quaternary diagrams recommended by the IMA for depiction of perovskite compositions. Up to 200 analyses can be entered into the spreadsheet, which is accompanied by data calculated for 140 perovskite compositions compiled from the literature.

1. Introduction Synthetic compounds and naturally-occurring minerals of the perovskite type adopt one of the most chemically-accommodating crystal structures known. Unlike many other structure types, nearly every element in the periodic table, including most of the noble gases and some organic molecules, can be found in some variant of the ideal perovskite structure. The name perowskin (perowskyn) was used initially for a lithium ironmanganese phosphate by Nils Gustaf Nordenski€ old in honour of Count Lev Alekseevich von Perovski, and was mentioned by J€ ons Jacob Berzelius in 1835 in his Jahres-Bericht and republished in Annalen der Physik (Berzelius, 1835a,b) under the name tetraphyllin. This latter name followed the usage of Johann Nepomuk Fuchs for lithium iron phosphate: triphyllin (Fuchs, 1834). Four years later, CaTiO3 perowskit (perovskite) was named by Gustav Rose – from material provided by A.B. Kaemmerer (Pekov, 1998) – also in honour of Lev Alekseevich von Perovski, and published in 1839, in both Annalen der Physik (in German), and printed separately in Latin by A.G. Schade (Rose, 1839a,b). Subsequent to the pioneering studies of Goldschmidt (1926a,b),

thousands of synthetic perovskites have been prepared (Galasso, 1990; Mitchell, 2002; Bruce et al., 2010; Vasala and Karppinen, 2015), and to date 43 naturally-occurring oxide, halide, hydroxide, arsenide and intermetallic compounds have been recognized in the perovskite supergroup of minerals (Mitchell et al., 2017). Compounds with the ideal perovskite structure, or its derivatives, have become some of the most important modern industrial materials, as they have an extremely wide variety of applications. Apart from their well-known (Galasso, 1990) electrical (ferroelectric, superconductivity) and magnetic properties (giant magnetoresistance), novel applications include: light-activated photocatalysis (Kanhere and Chen, 2014), perovskite-based solar cells (Petrus et al., 2017), solid oxide fuel cells (Sunarso et al., 2017), and radioactive waste sequestration (Lumpkin et al., 2014). Perovskites based upon the ABO3 structure (A ¼ Ca, Sr, Na, REE, Ba, Pb; B ¼ Ti, Zr, Nb, Sn, Si) are the most abundant of the naturallyoccurring species, with the silicate perovskite bridgmanite now considered to be the most abundant mineral in the Earth (Tschauner et al., 2014). Because of the extensive solid solutions possible at the A- and B-cation sites, natural oxide perovskites exhibit a wide range in

* Corresponding author. E-mail addresses: [email protected] (A.J. Locock), [email protected] (R.H. Mitchell). https://doi.org/10.1016/j.cageo.2018.01.012 Received 28 August 2017; Received in revised form 21 November 2017; Accepted 17 January 2018 Available online 31 January 2018 0098-3004/© 2018 Elsevier Ltd. All rights reserved.

A.J. Locock, R.H. Mitchell


Table 1 Examples of input data for perovskite analyses. Element

Na Mg Al Si K Ca Sc Ti Cr(III) Mn(II) Fe(III) Sr Zr Nb Sn(IV) La Ce(III) Pr Nd Sm Hf Ta Th U(VI) Total

Reference

Chakhmouradian et al. (2013)

Kopylova et al. (1997)

Galuskin et al. (2011)

Mitchell et al. (1998)

Table entry

Table 1.7

Table 2.413.1

Table 1.6

Table 1.7

Analyzed wt%

Analyzed wt%

0.02

2.90 0.88

Form

Analyzed wt%

Analyzed wt%

Na2O MgO Al2O3 SiO2 K2O CaO Sc2O3 TiO2 Cr2O3 MnO Fe2O3 SrO ZrO2 Nb2O5 SnO2 La2O3 Ce2O3 Pr2O3 Nd2O3 Sm2O3 HfO2 Ta2O5 ThO2 UO3

0.89

0.16

0.31

0.61

35.26

4.46 1.81

0.330

53.34

29.78 7.81

1.38 0.18 0.12 1.47

0.43a 19.95

1.13 3.01 0.24 1.09 0.12

17.46 12.69

3.51

29.26 0.23 2.31 0.16 0.55 0.16 21.95 0.11 43.80 0.09 0.11

28.29 17.51 0.26 10.09 0.24 35.23 1.11 2.36 0.24 0.71

0.65 0.39 1.07

1.44

0.11

100.00

100.11

0.41 0.24 100.38

99.93

For clarity, some vacant rows present in the spreadsheet have been omitted from this table. a Total iron; recalculated from FeO.

1997; Mitchell and Chakhmouradian, 1998; Mitchell et al., 1998; Chakhmouradian and Mitchell, 2001; Mitchell, 2002; Galuskin et al., 2008; Ma and Rossman, 2008; Bowles et al., 2011; Galuskin et al., 2011; Chakhmouradian et al., 2013; Galuskin et al., 2014; Khoury et al., 2015). As a result, compositional variations in specimens of these two subgroups are typically represented through the molar proportions of ideal end-members, and a series of ternary and quaternary diagrams is used to aid in their classification (Figs. 9, 13, 16 and 18 of Mitchell et al., 2017). Although the calculation of an empirical formula from a chemical analysis is a straightforward linear transformation, the determination of end-member proportions in these minerals is more complex; usually there are fewer constituent oxides (some or all of: Na2O, CaO, TiO2, Fe2O3, SrO, ZrO2, Nb2O5, SnO2, BaO, REE2O3, PbO, ThO2, UO3) than there are possible end-members (for example: NaNbO3, Na0.5REE0.5TiO3, BaTiO3, SrTiO3, REEFeO3, REETiO3.5, CaZrO3, CaSnO3, PbTiO3, Ca1.5U0.5O3, CaTiO3, Th0.25NbO3, CaNb0.5Fe0.5O3, CaNbO3.5, Th0.5TiO3). From the viewpoint of linear algebra, such a system is said to be underdetermined. As discussed by Rickwood (1968) for the silicate garnets (another underdetermined system), “two mineralogists can arrive at quite different compositions [proportions] by having calculated the molecules [end-members] in a different sequence.” Therefore, (and it can hardly be over-emphasized), a common scheme for the calculation of end-members is necessary in order to make valid comparisons of such proportions. Calculation of perovskite end-member proportions (for the anhydrous-oxide minerals) has previously been carried out using an APL program (Mitchell, 1996) that had limited availability. This work presents an Excel spreadsheet for such calculation, and takes advantage of recent implementations for Excel of triangular plotting (Graham and Midgley, 2000) and tetrahedral plotting (Shimura and Kemp, 2015) to automatically depict data in the appropriate compositional diagrams from Mitchell et al. (2017).

composition. This variation in composition, especially in the system tausonite-perovskite-lueshite-loparite (SrTiO3-CaTiO3-NaNbO3-REETi2O6), is used by petrologists to evaluate magmatic differentiation in a variety of igneous rocks (e.g., Mitchell and Vladykin, 1993; Mitchell and Chakhmouradian, 1996). To illustrate the compositional variation it is necessary to recalculate the oxide-based composition into potential end-member “molecules”. The sequence of calculation of the end-members is important if comparisons are to be made between different sets of analytical data. The establishment of a common recalculation scheme is the objective of this work. 2. Nomenclature The nomenclature of the perovskite supergroup – minerals that have, or that are derivative from, the aristotypic cubic perovskite crystal structure (Goldschmidt, 1926a; Lefkowitz et al., 1966) – has recently been revised and approved by the International Mineralogical Association (Mitchell et al., 2017). Consideration of the restrictions and variations that arise from the combination of crystal structure and composition has resulted in a hierarchical classification scheme, which for the stoichiometric-perovskite minerals results in three groups: the single perovskites (ABX3), the double perovskites (A2BB’X6), and the double antiperovskites (B2XX’A6): see Table 1 of Mitchell et al. (2017). The present work is concerned with the anhydrous-oxide (non-silicate) minerals, which include the perovskite subgroup of the group of stoichiometric single-perovskites, and the vapnikite subgroup of the group of stoichiometric double-perovskites. The silicate-perovskite bridgmanite is excluded from consideration, because of the restricted paragenesis of this mineral (Tschauner et al., 2014). Many of the mineral species of the perovskite and vapnikite subgroups (barioperovskite – BaTiO3, isolueshite – NaNbO3, lakargiite – CaZrO3, loparite Na0.5REE0.5TiO3, lueshite – NaNbO3, macedonite – PbTiO3, megawite – CaSnO3, perovskite – CaTiO3, and tausonite – SrTiO3; vapnikite – CaU0.5Ca0.5O3 and latrappite – CaNb0.5Fe0.5O3) show extensive solid solution (Radusinovic and Markov, 1971; Vorob'yev et al., 1984; Mitchell, 1996; Chakhmouradian et al., 1997; Kopylova et al.,

3. Spreadsheet description The Excel spreadsheet consists of four revealed worksheets: 107



Table 2 Examples of output for perovskite analyses.

Examples of output for perovskite analyses. Reference

Chakhmouradian et al. (2013)

Kopylova et al. (1997)

Galuskin et al. (2011)

Mitchell et al. (1998)

Table entry

Table 1.7

Table 2.413.1

Table 1.6

Table 1.7

0.4%

1.3%

3.7%

3.3%

15.31%

unassigned cations Normalized Endmembers (15)

Sequence and Formula per 1 B

lueshite

NaNbO3

1.83%

5.17%

0.00%

loparite

Na0.5REE0.5TiO3

4.54%

28.78%

0.00%

0.00%

REEFeO3

REEFeO3

2.57%

21.76%

0.24%

4.40%

barioperovskite

BaTiO3

0.00%

0.00%

0.00%

0.00%

tausonite

SrTiO3

0.25%

37.73%

0.31%

0.38%

REE2Ti2O7

REETiO3.5

0.00%

0.00%

0.00%

0.00%

lakargiite

CaZrO3

0.14%

0.00%

35.92%

0.00%

megawite

CaSnO3

0.00%

0.00%

57.61%

0.00%

macedonite

PbTiO3

0.00%

0.00%

0.00%

0.00%

vapnikite

Ca1.5U0.5O3

0.00%

0.00%

0.33%

0.00%

perovskite

CaTiO3

89.73%

6.32%

5.43%

35.49%

ThNb4O12

Th0.25NbO3

0.00%

0.00%

0.16%

0.00%

latrappite

CaNb0.5Fe0.5O3

0.00%

0.00%

0.00%

32.56%

Ca2Nb2O7

CaNbO3.5

0.00%

0.00%

0.00%

11.86%

ThTi2O6

Th0.5TiO3

0.94%

0.23%

0.00%

0.00%

Formula A B O3 Simplified A site

uses Na*

(Ca0.8946REE0.0483Na* 0.0409Th0.0058Sr0.0025) Σ0.9921

(Sr0.3716REE0.3561Na*0. 1927Ca0.0623Th0.0105)Σ0. 9932

(Ca1.0006Th0.003Sr0.003R EE0.0024)Σ1.009

(Ca0.7918Na*0.1469 REE0.0422Sr0.0036) Σ0.9845

Introduction worksheet that contains instructions and a short description of the spreadsheet; FORMULA worksheet that contains both the input compositional data and the output (proportions of the end-members, and simplified formula) for each analysis; FIGURES worksheet in which the data entered in the FORMULA worksheet are automatically plotted on the compositional diagrams (three ternary and one quaternary); Literature worksheet that tabulates the data and results for 140 analyses from the literature cited in the Introduction of this paper as well as data for 21 end-members;

compositions of these minerals are conventionally expressed as oxides in percent by weight, compositional data should be entered into the FORMULA worksheet (Table 1) as the following 36 oxides: Na2O, MgO, Al2O3, SiO2, K2O, CaO, Sc2O3, TiO2, Cr2O3, MnO, Fe2O3, SrO, Y2O3, ZrO2, Nb2O5, SnO2, BaO, La2O3, Ce2O3, Pr2O3, Nd2O3, Sm2O3, Eu2O3, Gd2O3, Tb2O3, Dy2O3, Ho2O3, Er2O3, Tm2O3, Yb2O3, Lu2O3, HfO2, Ta2O5, PbO, ThO2, UO3. Despite the existence of multiple oxidation states in the crust for such elements as Fe, Sn, Ce, Pb and U, the data should be entered as the oxides shown above, as these oxidation states match those of the endmembers. This spreadsheet has been adapted from a more generalized spreadsheet; as a result, and for display reasons, a large number of rows are hidden on the FORMULA worksheet. Up to 200 analyses can be entered (one per column) in the FORMULA worksheet, and for each analysis, a label should be assigned.

In addition to the revealed worksheets, there are five hidden worksheets; four of these provide the numerical transformations and calculations to enable automatic depiction of the data on the four diagrams, whereas the fifth is a utility worksheet that enables mass interconversions between different forms of the elements (e.g., Fe, FeO, and Fe2O3). They have been hidden to prevent accidental overwriting of the functions in these five worksheets. To display a hidden worksheet in Excel 2010, use the following steps: Home tab, Cells group, Format j Visibility j Hide & Unhide j Unhide Sheet j select the sheet of interest j OK.

3.2. Formula proportions The spreadsheet calculates the formula proportions on the basis of exactly three oxygen atoms per formula unit, and uses the input oxidation states (e.g., all of the iron is assumed to be ferric). The proportions of the cations are not fixed, both because of analytical uncertainty (propagated error) in the compositional data, and because of the possibility of vacancies in the perovskite crystal structure; however, the cation total should be close to two (preferably within less than one percent).

3.1. Data input Although the compositions of anhydrous-oxide (non-silicate) perovskite minerals are dominated by twelve elements (Na, Ca, Ti, Fe, Sr, Zr, Nb, Sn, Ba, Pb, Th, U) in addition to the lanthanides (La to Lu), there are several other elements that have been reported at minor to major concentrations, including: Mg, Al, Si, K, Sc, Cr, Mn, Y, Hf and Ta. As the

3.3. Cation assignments The cations are assigned to the perovskite formula (ABO3) as follows (Mitchell, 2002; Mitchell et al., 2017): 108



Table 3 The effect of calculation sequence on the end-member proportions, for ideal (Na1/3Ca1/3Ce1/3)

(Ti1/3Fe1/3Nb1/3)O3.

cation

Na

Ca

Ti

Fe

Nb

Ce

sum

percentage

apfu

0.333

0.333

0.333

0.333

0.333

0.333

2

100.00

0.667 0 0.667 0 0.667 0

33.33 0 33.33 0 33.33 0 0 0

original sequence lueshite NaNbO3 Loparite Na0.5REE0.5TiO3 REEFeO3 REEFeO3 REETiO3.5 REE2Ti2O7 perovskite CaTiO3 latrappite CaNb0.5Fe0.5O3 Ca2Nb2O7 CaNbO3.5 remainder switch order of Na members Loparite Na0.5REE0.5TiO3 lueshite NaNbO3 REEFeO3 REEFeO3 REE2Ti2O7 REETiO3.5 perovskite CaTiO3 latrappite CaNb0.5Fe0.5O3 CaNbO3.5 Ca2Nb2O7 remainder inverted sequence Ca2Nb2O7 CaNbO3.5 latrappite CaNb0.5Fe0.5O3 perovskite CaTiO3 REE2Ti2O7 REETiO3.5 REEFeO3 REEFeO3 loparite Na0.5REE0.5TiO3 lueshite NaNbO3 remainder

0.333

0.333 0.333

0

0.333

0.333

0

0

0.167 0.167

0

0.333 0.167

0.333 0

0

0.167

0.667 0.333 0.333 0 0 0.167

0

0

0.167

0.167

0.667

0

0

0.333

0

0

0.333

0.333

0.333

0

0.167

0

0.333

0.333

A: Na, Ca, K, Sr, Ba, Y, La to Lu, Pb, and Th. B: Mg, Al, Si, Sc, Ti, Cr, Mn, Fe, Zr, Nb, Sn, Hf, Ta, and U.

0

0

0

33.33 16.67 16.67 0 0 33.33 0 0

0.667 0 0 0.667 0 0 0 0.667

33.33 0 0 33.33 0 0 0 33.33

ending with Th0.5TiO3). This sequence was determined mainly from the requirement for the results to be in substantial agreement with those reported in Mitchell (1996, 2002), as well as the need to minimize the number and proportions of hypothetical end-members and the proportion of unattributed cations. The choice of end-members relies chiefly on the recent IMA nomenclature (Mitchell et al., 2017), as well as earlier literature (Mitchell, 1996,; Kopylova et al., 1997; Mitchell et al., 1998; Mitchell, 2002). The oxidation states of the elements have been chosen to match those of the end-members used, e.g., Sn(IV), Ce(III), Pb(II), and U(VI), and Fe(III), as supported by M€ ossbauer spectroscopy for latrappite (Mitchell et al., 1998). The proportions of the 15 end-members are calculated by successive deduction from a formula that is based on three oxygen atoms, and thus the calculation sequence is independent of the starting composition. Although the spreadsheet is intended for use with the anhydrous-oxide minerals of the groups of stoichiometric single- and double-perovskites, some of the end-members used are actually non-stoichiometric (e.g., REETiO3.5, Th0.25NbO3, CaNbO3.5, Th0.5TiO3). However, the compositions of many natural perovskite samples require such end-members for their description (Mitchell, 1996; Mitchell et al., 1998). Any user of this spreadsheet must recognize the effects of the grouping of elements (described above in section 3.3 Cation assignments) on the calculation of the end-member proportions. In particular, the lanthanide elements and yttrium are grouped together as REE, and sodium and potassium are grouped together. Such grouping can give rise to results that may not be expected by the unwary reader – for example, the spreadsheet reports substantial lueshite (ideally NaNbO3) and loparite (ideally Na0.5REE0.5TiO3) for the analysis of Kopylova et al. (1997) that is nearly devoid of sodium (0.16 wt% Na2O), but which has substantial potassium (4.46 wt% K2O). In this case, it should be clear that the end-members reported are in fact the potassium analogues of lueshite and loparite (Table 2). In most analyses of natural anhydrous-oxide perovskite minerals, calculation of end-member proportions will result in a certain amount of cations that is not assigned to any of the 15 end-members listed in Table 2. In the current spreadsheet, the unassigned cations are expressed as a percentage of the total cations calculated on the formula basis of

This inflexible attribution of cations leads, in the case of vapnikite, to the irregular formula (Ca)1.5(U)0.5O3, rather than the correct (Ca)(U0.5Ca0.5)O3. However, this irregularity is one of expression only, and is perhaps mitigated by the rarity of vapnikite coupled with its lack of significant solid solution (Galuskin et al., 2014; Khoury et al., 2015). For simplicity (and to reduce the proliferation of theoretical endmember components), several of the elements are grouped together, both in the presentation of the simplified formula, and more importantly, during calculation of the end-member proportions. These groupings are denoted in the formulas with an asterisk (with the exception of the lanthanides), and include:

0

Na* ¼ Na þ K; Fe* ¼ Fe þ Al þ Cr þ Sc; Zr* ¼ Zr þ Hf; Nb* ¼ Nb þ Ta; REE ¼ La to Lu þ Y.

The groupings are based on similarities in the geochemical behaviors of the elements, along with the relative rarity of occurrence of some of the elements in anhydrous-oxide perovskite minerals (e.g., K, Al, Cr, and Sc).

3.4. Sequential calculation of end-member proportions End-member calculation schemes have the general problem that “the calculated composition depends on the sequence of assignment of cations to particular end members” (Mitchell, 1996). This was discussed in some detail for silicate garnets by Rickwood (1968) who emphasized the need for a standard method of calculation. Mitchell (1996) suggested that this problem could be somewhat “reduced if the end members are formed from the least to the most abundant”. In the spreadsheet described here, the proportions of 15 end-members are generated by sequential calculation in the fixed order shown in Table 2 (starting with NaNbO3 and 109



Fig. 1. Compositional fields (mol%) for the ternary system that includes the end-members tausonite – perovskite – loparite, with recommended subdivisions (Mitchell et al., 2017). The data presented in Tables 1 and 2 correspond to the points: 1 – Chakhmouradian et al. (2013); 2 – Kopylova et al. (1997); 3 – Galuskin et al. (2011); 4 – Mitchell et al. (1998). The ternary diagram is plotted with the method of Graham and Midgley (2000).

3.5. Data output

three oxygen atoms. Such unassigned cations include both those elements that are not assigned to end-members at all in this spreadsheet (Mg, Si, Mn), as well as any cations that remain after sequential calculation of the 15 end-members. Examination of a set of 140 analyses of perovskite (tabulated in the Literature worksheet) yields a median value for unassigned cations of 2.0%, an average value of 3.0%, and a value of 5% at the 75th percentile. There are several possible explanations for an analysis that has more than 5% unassigned cations:

Table 2 presents the results from the spreadsheet for the four analyses whose input data are listed in Table 1. For each analysis, the percentage of unassigned cations is presented below the analytical total in the FORMULA worksheet; values in excess of 5% are automatically highlighted with a light-red fill and dark-red text using the Excel style of Conditional Formatting. The proportions of the 15 perovskite end-members are normalized to the end-members present (that is, without the unassigned cations) and are expressed as percentages (listed to two decimal places below the chemical analysis with which they correspond). Conditional Formatting (graded three-color scale with the midpoint at the 80th percentile) is also used to draw attention to the most abundant end-members (the intensity of the yellow shading increases with abundance). The end-member proportions can be converted easily by the user to raw proportions (for which the unassigned cations make up the difference) as follows:

1) the analysis may contain some uncommon end-member (e.g., Sr2Nb2O7, or Ca2Fe2O5) that is not part of the list of 15 used in this work, 2) the analysis may contain some element not assigned to an endmember (Mg, Si, Mn), 3) the analysis may be of inferior quality; analytical error propagates into the calculation of end-member proportions. As stated above, the sequence of calculation of the end-members affects the results. Table 3 shows the effects of different calculation sequences for the ideal formula (Na1/3Ca1/3Ce1/3) (Ti1/3Fe1/3Nb1/3)O3. The seven end-members considered in Table 3 are those in the system Na-Ca-REETi-Fe-Nb-O. For the original sequence of calculation (the sequence analogous to that used in Table 2 and in the spreadsheet), the results are: lueshite 33.33%, REEFeO3 33.33%, and perovskite 33.33%. However, if just the order of the two Na-bearing end-members is reversed, the results change considerably; this revised sequence of calculation yields: loparite 33.33%, lueshite 16.67%, REEFeO3 33.33%, and latrappite 33.33%. Similarly, if the original sequence is inverted, the calculation gives: CaNbO3.5 33.33%, REETiO3.5 33.33%, remainder (unassigned) 33.33%. The substantial influence on the results of the order-of-calculation of the end-members requires that a single uniform sequence be adopted for such calculations. Otherwise, comparison of results will not be meaningful. For this reason, the sequence adopted in this manuscript and spreadsheet – derived from comparison to Mitchell (1996, 2002) and the recent IMA nomenclature (Mitchell et al., 2017) – is fixed, and not intended to be editable.

Raw value ¼ Normalized value * [(100 – percentage of unassigned cations)/100]. The simplified formula for each analysis is presented below the endmember proportions both by site (A and B) and concatenated (ABO3) using, where necessary, the grouped elements: Na*, Fe*, Zr*, Nb*, and REE (see section 3.3 Cation assignments). These are followed by the individual elements and their abundances (if present) for each element grouping, in order of decreasing abundance of the constituent elements (with the exception of the lanthanides, which are given in order of increasing atomic number, followed by yttrium). As a result of the limitations of automatic text generation in Excel, numerical quantities are not subscripted in these formulas.

3.6. Compositional diagrams The compositional diagrams used in Mitchell et al. (2017) – specifically their Figs. 9, 13, 16 and 18 – are reproduced on the FIGURES worksheet (in Arial font) as the Tausonite - Perovskite - Loparite diagram, the Lueshite - Perovskite - Loparite diagram, the Perovskite - Megawite 110



Fig. 2. Compositional fields (mol%) for the ternary system that includes the end-members lueshite – perovskite – loparite, with recommended subdivisions (Mitchell et al., 2017). The data presented in Tables 1 and 2 correspond to the points: 1 – Chakhmouradian et al. (2013); 2 – Kopylova et al. (1997); 3 – Galuskin et al. (2011); 4 – Mitchell et al. (1998). The ternary diagram is plotted with the method of Graham and Midgley (2000).

Fig. 3. Compositional fields (mol%) for the ternary system that includes the end-members perovskite – megawite – lakargiite (Mitchell et al., 2017). The data presented in Tables 1 and 2 correspond to the points: 1 – Chakhmouradian et al. (2013); 2 – Kopylova et al. (1997); 3 – Galuskin et al. (2011); 4 – Mitchell et al. (1998). The ternary diagram is plotted with the method of Graham and Midgley (2000).

including ease of customization. The original Tri-plot spreadsheet of Graham and Midgley (2000) can be found at: http://www.lboro.ac.uk/ microsites/research/phys-geog/tri-plot/. The Tetrahedral plot diagram spreadsheet of Shimura and Kemp (2015) is given as an appendix to their manuscript, American Mineralogist deposit item# AM-15-115371, at: http://www.minsocam.org/msa/ammin/toc/2015/ND2015_data/ ND2015_data.html.

Lakargiite diagram, and the Lueshite - Perovskite - Latrappite - Ca2Nb2O7 diagram, respectively. The spreadsheet takes advantage of recent implementations for Excel of triangular plotting (Graham and Midgley, 2000) and tetrahedral plotting (Shimura and Kemp, 2015) to automatically depict all of the data that have been entered in the FORMULA worksheet. The triangular and tetrahedral plotting procedures that have been implemented in the current spreadsheet were chosen for their ease of use,

111



Fig. 4. a. Compositional fields (mol%) for the quaternary system lueshite – perovskite – latrappite – CaNbO3.5 (Mitchell et al., 2017) plotted at gamma of 60 and dip of 35 (after Shimura and Kemp, 2015). The data presented in Tables 1 and 2 correspond to the points:1 – Chakhmouradian et al. (2013); 2 – Kopylova et al. (1997); 3 – Galuskin et al. (2011); 4 – Mitchell et al. (1998). b. The diagram plotted at gamma of 75 and dip of 15 (after Shimura and Kemp, 2015).

a

b

112



Appendix A. Supplementary data

The four analyses whose end-member proportions are listed Table 2 consist primarily of perovskite (results column #1, from Chakhmouradian et al., 2013); tausonite, loparite and REEFeO3 (results column #2, from Kopylova et al., 1997); megawite and lakargiite (results column #3, from Galuskin et al., 2011); and perovskite, latrappite, lueshite, and Ca2Nb2O7 (results column #4, from Mitchell et al., 1998). These analyses are shown as labelled points on the Tausonite - Perovskite - Loparite diagram in Fig. 1. Note that the use of ternary or quaternary diagrams, as pointed out by Chayes (1971), involves “the reduction of some subset of the original variables to a new set of proportions”; such diagrams are projections from multivariate-sample space into two- or three-dimensional space. An unintended consequence of this dimensional reduction is that the resultant depiction of the data may mislead the unsuspecting user. Thus, Fig. 1 shows that three of the four analyses fall very close to the perovskite apex; however, inspection of Table 2 reveals that only analysis #1 has more than 50% of the perovskite end-member. In a similar fashion, although on Fig. 1 analysis #2 falls into the tausonite field, on Fig. 2 the same analysis falls into the loparite field, whereas in Table 2 it is shown the end-member abundances decrease in the order: tausonite > loparite > REEFeO3. Thus, it is crucial not to rely on the figures in isolation, but to also examine the full set of (tabulated) end-member proportions. Fig. 3 illustrates the Perovskite - Megawite Lakargiite diagram; note that analysis #3 is well inside the megawite field, and not near the perovskite apex (its location in Figs. 1, 2 and 4). Fig. 4a shows the Lueshite - Perovskite - Latrappite - Ca2Nb2O7 tetrahedron in the same orientation as that given in Fig. 18 of Mitchell et al. (2017): at rotation angle gamma of 60 and dip (angle of the observer) of 35 , following the nomenclature used in the appendix of Shimura and Kemp (2015). Fig. 4b shows the same tetrahedron and data but at gamma of 75 and dip of 15 . The ability of the spreadsheet user to control the orientation of this quaternary diagram can provide valuable different perspectives for the same data.

Supplementary data related to this article can be found at https://doi. org/10.1016/j.cageo.2018.01.012. References Berzelius, J., 1835a. Mineralogie. Neue Mineralien. Tetraphyllin. Jahres-Bericht über die Fortschritte der physischen Wissenschaften, vol. 15, pp. 211–213. Berzelius, J., 1835b. Triphyllin und Tetraphyllin, verwandte Mineralien. Ann. Phys. 112 (previously 36) (11), 473–475. Bowles, J.F.W., Howie, R.A., Vaughan, D.J., Zussman, J., 2011. Rock-forming Minerals Volume 5A. Non-silicates: Oxides, Hydroxides, Sulphides, second ed. Geological Society of London. 920 p. Bruce, D.W., O'Hare, D., Walton, R.I. (Eds.), 2010. Functional Oxides. John Wiley & Sons, Chichester, 304 p. Chakhmouradian, A.R., Mitchell, R.H., 2001. Three compositional varieties of perovskite from kimberlites of the Lac de Gras field (Northwest Territories, Canada). Mineral. Mag. 65, 133–148. Chakhmouradian, A.R., Reguir, E.P., Kamenetsky, V.S., Sharygin, V.V., Golovin, A.V., 2013. Trace-element partitioning in perovskite: implications for the geochemistry of kimberlites, other mantle-derived undersaturated rocks. Chem. Geol. 353, 112–131. Chakhmouradian, A., Yakovenchuk, V., Mitchell, R.H., Bogdanova, A., 1997. Isolueshite: a new mineral of the perovskite group from the Khibina alkaline complex. Eur. J. Mineral 9, 483–490. Chayes, F., 1971. Ratio Correlation: A Manual for Students of Petrology and Geochemistry. University of Chicago Press, Chicago, 99 p. Fuchs, J.N., 1834. Ueber ein neues Mineral (Triphylin). J. Prakt. Chem. 3, 98–104. Galasso, F.S., 1990. Perovskites and High-Tc Superconductors. Gordon and Breach Science Publishers, New York, 294 p. Galuskin, E.V., Galuskina, I.O., Gazeev, V.M., Dzierzanowski, P., Prusik, K., Pertsev, N.N., Zadov, A.E., Bailau, R., Gurbanov, A.G., 2011. Megawite, CaSnO3: a new perovskitegroup mineral from skarns of the upper chegem caldera, Kabardino-Balkaria, northern caucasus, Russia. Mineral. Mag. 75, 2563–2572. Galuskin, E.V., Galuskina, I.O., Kusz, J., Armbruster, T., Marzec, K.M., Dzier_zanowski, P., Murashko, M., 2014. Vapnikite Ca3UO6 – a new double-perovskite mineral from pyrometamorphic larnite rocks of the Jabel Harmun, Palestinian Autonomy, Israel. Mineral. Mag. 78, 571–582. Galuskin, E.V., Gazeev, V.M., Armbruster, T., Zadov, A.E., Galuskina, I.O., Pertsev, N.N., Dzier_zanowski, P., Kadiyski, M., Gurbanov, A.G., Wrzalik, R., Winiarski, A., 2008. Lakargiite CaZrO3: a new mineral of the perovskite group from the north caucasus, Kabardino-Balkaria, Russia. Am. Mineral. 93, 1903–1910. Goldschmidt, V.M., 1926a. Geochemische Verteilungsgesetze VII. Die Gesetze der Krystallochemie nach Untersuchungen gemeinsam mit T. Barth, G. Lunde, W. Zacharisasen. Skrifter utgitt av det Norske Videnskaps-Akademi i Oslo 1: MatematiskNaturvidenskapelig Klasse, pp. 1–117. Goldschmidt, V.M., 1926b. Die Gesetze der Krystallochemie. Naturwissenschaften 14, 477–485. Graham, D.J., Midgley, N.G., 2000. Graphical representation of particle shape using triangular diagrams: an Excel spreadsheet method. Earth Surf. Process. Landforms 25, 1473–1477. Kanhere, P., Chen, Z., 2014. A review on visible light active perovskite-based photocatalysts. Molecules 19, 19995–20022. Khoury, H.N., Sokol, E.V., Clark, I.D., 2015. Calcium uranium oxide minerals from Central Jordan: assemblages, chemistry, and alteration products. Can. Mineral. 53, 61–82. Kopylova, M.G., Gurney, J.J., Daniels, L.R., 1997. Mineral inclusions in diamonds from the River Ranch kimberlite, Zimbabwe. Contrib. Mineral. Petrol. 129, 366–384. Lefkowitz, I., Łukaszewicz, K., Megaw, H.D., 1966. The high-temperature phases of sodium niobate and the nature of transitions in pseudosymmetric structures. Acta Crystallogr. 20, 670–683. Lumpkin, G.R., Gao, Y., Giere, R., Williams, C.T., Mariano, A.N., Geisler, T., 2014. The role of Th-U minerals in assessing the performance of nuclear waste forms. Mineral. Mag. 78, 1071–1095. Ma, C., Rossman, G.R., 2008. Barioperovskite, BaTiO3, a new mineral from the benitoite mine, California. Am. Mineral. 93, 154–157. Mitchell, R.H., 1996. Perovskites: a revised classification scheme for an important rare earth element host in alkaline rocks. In: Jones, A.P., Wall, F., Williams, C.T. (Eds.), Rare Earth Minerals: Chemistry, Origin and Ore Deposits. Chapman & Hall, London, pp. 41–76. Mitchell, R.H., 2002. Perovskites: Modern and Ancient. Almaz Press, Thunder Bay, 318 p. www.almazpress.com. Mitchell, R.H., Chakhmouradian, A.R., 1996. Compositional variation of loparite from the Lovozero alkaline complex. Can. Mineral. 34, 977–990. Mitchell, R.H., Chakhmouradian, A.R., 1998. Th-rich loparite from the Khibina alkaline complex, Kola Peninsula: isomorphism, paragenesis. Mineral. Mag. 62, 341–353. Mitchell, R.H., Vladykin, N.V., 1993. Rare earth element-bearing tausonite and potassium barium titanates from the Little Murun potassic alkaline complex, Yakutia, Russia. Mineral. Mag. 57, 651–664. Mitchell, R.H., Choi, J.B., Hawthorne, F.C., McCammon, C.A., Burns, P.C., 1998. Latrappite; a re-investigation. Can. Mineral. 36, 107–116. Mitchell, R.H., Welch, M.D., Chakhmouradian, A.R., 2017. Nomenclature of the perovskite supergroup: a hierarchical system of classification based on crystal structure, composition. Mineral. Mag. 81, 411–461.

4. Testing of the spreadsheet From the references cited in section 2 above, 140 analyses were compiled in the Literature worksheet along with idealized data for 21 theoretical end-member compositions; this dataset was used to test the utility and validity of the spreadsheet. In particular, the selection of endmembers and the order (sequence) of their calculation was optimized by comparing the results tabulated in the Literature worksheet with the original references, most notably Mitchell (1996, 2002). This compilation is not meant to be comprehensive, and is restricted to anhydrous oxide (non-silicate) perovskite supergroup minerals (the majority of which occur in alkaline rocks or were derived from upper mantle rocks), and notably omits bismuth, as this element appears in abundance only in macedonite (Radusinovic and Markov, 1971). 5. System requirements and program availability The spreadsheet has been tested with Excel 2010 using the Windows 7 operating system and with Microsoft Excel for Mac 2011 using the Mac OS X (V10.5.6) operating system. The spreadsheet is available as Supplementary material, or from the first author. Disclaimer This publication is not affiliated with, nor has it been authorized, sponsored, or otherwise approved by Microsoft Corporation or by Apple Inc.; Microsoft, Excel, and Windows are trademarks of Microsoft Corporation, and Mac OS is a trademark of Apple Inc. Acknowledgements The constructive comments of the reviewers are gratefully acknowledged. 113


Computers and Geosciences 113 (2018) 106–114 Shimura, T., Kemp, A.I.S., 2015. Tetrahedral plot diagram: a geometrical solution for quaternary systems. Am. Mineral. 100, 2545–2547. Sunarso, J., Hashim, S.S., Zhu, N., Zhou, W., 2017. Perovskite oxides applications in high temperature oxygen separation, solid oxide fuel cell and membrane reactor: a review. Prog. Energy Combust. Sci. 61, 57–77. Tschauner, O., Ma, C., Beckett, J.R., Prescher, C., Prakapenka, V.B., Rossman, G.R., 2014. Discovery of bridgmanite, the most abundant mineral in Earth, in a shocked meteorite. Science 346 (6213), 1100–1102. Vasala, S., Karppinen, M., 2015. A2B0 B00 O6 perovskites: a review. Prog. Solid State Chem. 43, 1–36. Vorob'yev, Y.I., Konev, A.A., Malyshonok, Y.V., Afonina, G.F., Sapozhnikov, A.N., 1984. Tausonite, SrTiO3, a new mineral of the perovskite group. Int. Geol. Rev. 26, 462–465.

Pekov, I.V., 1998. Minerals First Discovered on the Territory of the Former Soviet Union. Ocean Pictures Ltd., Moscow, 369 p. Petrus, M.L., Schlipf, J., Li, C., Gujar, T.P., Giesbrecht, N., Müller-Buschbaum, P., Thelakkat, M., Bein, T., Hüttner, S., Docampo, P., 2017. Capturing the Sun: a review of the challenges and perspectives of perovskite solar cells. Adv. Eng. Mater. 7, 1700264 (27 p). Radusinovic, D., Markov, C., 1971. Macedonite – lead titanite: a new mineral. Am. Mineral. 56, 387–394. Rickwood, P.C., 1968. On recasting analyses of garnet into end-member molecules. Contrib. Mineral. Petrol. 18, 175–198. Rose, G., 1839a. Beschreihung einiger neuen Mineralien des Urals. Der Perowskit, eine neue Mineralgattung. Ann. Phys. 124 (previously 48) (12), 551–573 (558-561). Rose, G., 1839b. De perowskite, fossili novo. In: A.G. Schade (Ed.), De Novis Quibusdam Fossilibus Quae in Montibus Uraliis Inveniuntur, pp. 3–5. Berlin, 12 p.

114